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In Landau Mechanics (third edition page 4), why does adding Lagrangians of two non interacting parts remove the indefiniteness of multiplying each Lagrangian by a different constant?

If both systems are completely non-interacting, I can perfectly create a new Lagrangian L = a*L1 + b*L2 and still maintain validity of the Euler-Lagrange equations and make the action integral perfectly stationary, with a and b different. Can't I?

Here's the original statement:

It is evident that the multiplication of the Lagrangian of a mechanical system by an arbitrary constant has no effect on the equations of motion. From this, it might seem, the following property of arbitrariness can be deduced: the Lagrangians of different isolated mechanical systems may be multiplied by different arbitrary constants. The additive property, however, removes this indefiniteness, since it admits only the simultaneous multiplication of the Lagrangians of all the systems by the same constant.

I kind of understand that, of course, after you create L = L1 + L2, xL = xL1 + xL2 but nothing stops me from multiplying each Lagrangian of each non interacting part by different constants a and b and then summing, like: xL = xaL1 + xbL2

So the statement seems without meaning for me. Could anyone clarify? Thanks!

One addition: I see that, if both systems are interacting externally, the constants must be equal (and the proportion of masses becomes relevant). My goal is exactly that: to understand, conceptually, how he derives the role of ration of masses later on in page 7. Going back to the text, what confused me is that the whole thing starts with the assumption that both system are closed so any external interaction is dismissed:

Let a mechanical system consist of two parts which, if closed (..), and the interaction between them may be neglected, the Lagrangian of the whole system tends to: lim L = La + Lb.

Gilberto
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3 Answers3

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The implicit assumption in Landau and Lifschitz is that you can interact with the systems externally, to add or remove energy. The energy/momentum of the system is determined from the Lagrangian, and removing energy from one system, and adding the same energy to the other will reveal the proportionality constant between the two.

The reason this is left implicit is because the Lagrangian formulation already has an implicit energy law inside, since the Lagrangian is constructed from energy concepts (although technically speaking it is logically independent, with energy conservation only following for a time independent system). So when you have the sum Lagrangian, you know the ratio of kinetic and potential energies in the two systems, something which is physically relating the two coefficients in front to each other.

Ron Maimon
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Since the Euler-Lagrange equations are homogeneous we can multiply the entire Lagrangian by a constant (or add a constant, incidentally) and not change the equations of motion. Not so when we attempt to multiply separate terms of the Lagrangian by different constants. Your equations of motion will not reduce to Newton's law, $\mathbf{F} = m \mathbf{a}$, among other problems (total energy will be wrong, angular momentum, etc.).

As a test case you can consider two particles in an external potential well (without mutual interactions). If you have your $a$ and $b$ above such that $a\ne b$, you'll deduce an equation of motion that's wrong.

MarkWayne
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First, consider the two systems to be close together so that there is are one or more coupling terms in the joint Lagrangian containing products of generalized coordinates (or velocities) for both systems. As the two systems are moved apart so that they do not interact, these coupling terms will vanish. In order to be consistent with the limiting process, only one arbitrary constant factor for the original Lagrangian is allowed. When the two systems are separated, the remaining joint Lagrangian is the sum of two independent Lagrangians multiplied by the original (arbitrary) factor. Following this line of reasoning, the observation of Landau is intimately related to the fact that two separate systems have a possibility of interacting with each other (if they are, for example, moved into close proximity with one another).

Tane Ray
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